Newton's Method is an application of Taylor Polynomials for finding roots of functions. In general, solving an equation ??(??) = 0 is not easy, though we can do it in simple cases like find roots of quadratics. If the function is complicated, we can approximate the solution using an iterative procedure also known as a numerical method. One simple method is called Newton's Method.

Suppose that ?? = ?? is an (unknown) zero of ??(x) and that ??(??) is differentiable in an open interval that contains c: To approximate c

1. Make an initial approximation ??1 close to ??.

2. Determine a new approximation using the formula

????+1 = ???? ? ??(????) ???(???? )

3 If |????+1 ? ????| is less than the desired accuracy (which will be specified), let ????+1 serve as the final approximation. Otherwise, return to step 2 and calculate a new approximation.

Each calculation of a successive approximation is called iteration.

I am suppose to be solving the equation or finding the zero for f(x)=x^7-1000 using Newtons method.

The thing is I dont know whether to use for loop or whole loop or how to set my initial sum and initial value. The initial value for me has to be 3. I also dont know how to formulate this break rule that if the new approximated value is a less than 0.0001 difference from the old approximated value than I have to break the loop. I also dont know what my control statement would be.

The written work is all I have so far. My program is whats inside the box.

Problem 1: Newton's Method Newton's Method is an application of Taylor Polynomials for finding roots of functions. In general, solving an equation f(x)0 is not easy, though we can do it in simple cases like find roots of quadratics. If the function is complicated, we can approximate the solution using an iterative procedure also known as a numerical method. One simple method is called Newton's Method. Suppose that c is an (unknown) zero off(x) and that f(x) is differentiable in an open interval that contains c: To approximate c 1. 2. Make an initial approximation x1 close to c Determine a new approximation using the formula If lxnt-Xnl is less than the desired accuracy (which will be specified), let xn+1 serve as the final approximation. Otherwise, return to step 2 and calculate a new approximation. Each calculation of a successive approximation is called iteration. Example: To use three iterations of Newton's Method to approximate a zero off(x)-x2-2: where x 1 is the initial guess. We need to know f'(x) 2x; and we now can use the formula (we are given that Xo 1) f (xo) = 1 =1+-=1.5 (x1) 1.5- 5 2(1.5) = 1.5-(2.25-2)/3 = 1.42 (1.42)2-2 2 (1.42) f (x2) - 1.41 (x)= 1.42-0429 Notice that the answer is getting closer to the correct answer x2 - 2 0 which gives x2- 2 x- V2 1.41 Now, write a MATLAB program that uses Newton's Method to approximate a zero of f(x) -x7- 1000: where x3 is the initial guess. Use Newton's Method and continue until two successive approximations differ by less than 0.0001. Run your program and determine the value of zero of f(x). Wha MATLAB program and output ? Hand in your